Difference between Ctrl P > Follow Path and Add Object Constraint
CtrlP > Follow Path parents the selected objects to the active object (=curve), enables Path Animation in the curve's Object data tab in the Properties Editor and adds an F-Curve for the Evaluation Time property. An F-Curve Modifier is added to that F-Curve of type Generator (first order expanded polynomial).

Adding a Follow Path constraint to an object and target it at a curve only relocates the object (apparently to one of the end points of the curve), you need to click Animate Path to add an F-Curve with modifier (1st order exp. polynomial generator again) to the curve object in order to let the object move along the curve.

If you don't want that offset when using a Follow Path constraint, select your object and a curve,
CtrlP > Path Constraint. It adds a Follow Path constraint, creates the F-Curve with modifier for the the curve and does not translate your object.